Suppose the sun expands so that its radius becomes 100 times its present radius and its surface temperature becomes half of its present value. The total en ergy emitted by it then will increase by a factor of

According to Stefan's law, the radiation energy emitted per second per unit area of a black body $=\sigma T^4$, where $\sigma$ is called Stefan's constant. Energy emitted $\propto A T^4 \Rightarrow E \propto 4 \pi r^2 T^4$ where $A=$ area of the body, $T=$ temperature in Kelvin. When sun expands such that its radius becomes 100 times its present radius and its surface temperature becomes half. Then energy emitted by it is $$ \begin{aligned} E^{\prime} & \propto 4 \pi(100)^2 r^2\left(\frac{T}{2}\right)^4 \\ & \propto\left(\frac{100}{4}\right)^2 E \end{aligned} $$ Total energy emitted will be increased by $$ \frac{E^{\prime}}{E}=\frac{(100 / 4)^2 E}{E}=625 . $$